IC/96/60

United Nations Educational, Scientific and Cultural Organizationand

International Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

FUNCTIONAL INTEGRALSFOR NUCLEAR MANY-PARTICLE SYSTEMS

Yu. Yu. Lobanov1

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

The new method for computation of the physical characteristics of quantum systemswith many degrees of freedom is described. This method is based on the representationof the matrix element of the evolution operator in Euclidean metrics in a form of thefunctional integral with a certain measure in the corresponding space and on the use ofapproximation formulas which we constructed for this kind of integrals. The method doesnot require preliminary discretization of space and time and allows to use the deterministicalgorithms. This approach proved to have important advantages over the other knownmethods, including the higher efficiency of computations. Examples of application of themethod to the numerical study of some potential nuclear models as well as comparisonof results with the experimental data and with the values obtained by the other authorsare presented.

MIRAMARE-TRIESTE

April 1996

Permanent address: Joint Institute for Nuclear Research, Dubna, Moscow Region,141980, Russian Federation.

1 Introduction

Investigation of quantum systems consisting of many interacting particles is one of thefundamental problems in physics [1]. The convenient framework for numerical studyof the systems with many degrees of freedom is a path-integral approach [2]. Beingone of the most important mathematical techniques for solution of the wide class ofproblems in computational physics [3], path integrals provide a useful tool for study of thevariety of nuclear systems which are otherwise not amenable to definitive analysis throughperturbative, variational or stationary-phase approximations, etc. (see [4]). However,the existing approaches to path integrals in physics are not always quite correct in amathematical sense and the usual method of their computation is Monte Carlo, whichgives the results only as probabilistic averages and requires too much computer resourcesto obtain the good statistics. Moreover, when the nuclear many-body problem is beingformulated on a lattice approximating the kinetic energy operator and the potential ofinteraction, the description of heavy nucleus would require the lattice size which is fourto five orders of magnitude more than any lattice gauge calculation and which is hardlypossible on any foreseable computer [4]. The existing results of computation of the bindingenergy even for the light nuclei by means of variational or Green function Monte Carlomethod, as well as by the coupled cluster and Faddeev equation methods differ one fromthe other and from the experimental values more than the estimated errors of calculation(see [5],[6]) while requiring several hours on Cray-YMP and Cray-2 computers [5]. It isclear that the creation of the new methods for solution of such a complicated problemis of high importance. Increasing attention is being paid nowdays to the construction ofdeterministic algorithms for computation of path integrals which would be more effectivethan conventional Monte Carlo (see [7]-[9]). For instance, in [7] the authors discuss themethod of path summation based on the introduction of the "short-time" propagatorin the range 0 < t < e, which is correct up to O(e2) and on the successive iterationsto the finite imaginary time t in order to obtain the ground state energy of a quantumsystem. It should be noted that the main physical properties such as energy spectrum,wave functions squared, etc. can be reproduced correctly only in a limit t —> oo. Inpaper [8] and in extending its results paper [9] the method of evaluation of discretizedpath integrals is developed on the basis of iteration of the short-time approximation andon the direct matrix multiplication. The authors state that this approach is more alongthe original lines of Feynman of integrating over all space at each intermediate timerather than accepting or rejecting of paths according to a Markov process in Monte Carloalgorithms. However, their method also assumes the preliminary discretization and itcan be applied to the systems with only a few quantum degrees of freedom. It is alsolimited by the size of the matrix that can be handled on existing computers (see [8]).It means that using this approach it is dificult to treat problems in which the systemis delocalized over large distances. The method reported in [8] and [9] contains severaladjustable parameters which affect the accuracy of results. The choice of these parametersis not clear and sometimes the decrease of spacing (which corresponds to approaching thecontinuum limit) gives worse result and requires more iterations to improve it (see [9]).

According to the recent achievements in the measure theory and in the developmentof functional integration methods (see, e.g. [10]) as well as in the functional methodsin quantum physics [11] one can create the mathematically well-grounded methods forcomputation of path integrals. One of the promising approaches is the construction of

approximation formulas which are exact on a given class of functionals [10]. Based onthe rigorous definition of a functional integral in complete separable metric space in theframework of this approach we elaborated the new numerical method of computationof path integrals [12]. This method does not require preliminary discretization of spaceand time and allows to obtain the mathematically well-grounded physical results with aguaranteed (not probabilistic) error estimate. It permits the straightforward computationof functional integrals over all space without any simplifying assumption (perturbationexpansion, semiclassical or short-time approximation, etc). Our approximation formulascan be interpreted as quadratures in functional spaces. They are exact on a class ofpolynomial functionals of a given degree. Our method contains therefore all advantagesof deterministic approach and is free from the limitations of other methods mentionedabove. Under determined conditions we have proved the convergence of approximationsto the exact value of the integral and estimated the remainder which gives upper andlower bounds of the result. We have studied in detail the functional measure of the Gaus-sian type and some of its important particular cases such as conditional Wiener measurein the space of continuous functions [13] in Euclidean quantum mechanics (or quantumstatistical mechanics) and the functional measure in a Schwartz distribution space in two-dimensional Euclidean quantum field theory [14]. Numerical computations show [14] thatour method gives significant (by an order of magnitude) economy of computer time andmemory versus conventional Monte Carlo method used by other authors in the problemswhich we have considered. This approach is also proven to have advantages in the caseof high dimensions when the other deterministic methods loose their efficiency [15]. Ourdeterministic algorithm of computation of functional integrals which we use in quantumstatistical physics instead of traditional stochastic methods enable us to make a construc-tive proof of existence of continuum limit of the lattice (discretized) path integrals and tocompute the physical quantities in this limit within the determined error bars. Using thismethod we performed the numerical investigation of topological susceptibility in quantumstatistical mechanics and computed the #-vacua energy in continuum for the first time[14]. Our algorithm realized in a program written in FORTRAN has been implementedon CDC-6500 and Convex C220 computers. However, it is also possible to use personalcomputers since the method does not require too much computer resources.

In the present paper we discuss the mathematical foundations of the method andits numerical aspects. We extend it to the case of fermions (i.e. the anticommutingvariables) in continuum limit and apply to the study of some potential nuclear models.The comparison of our numerical results with the experimental data and with the valuesobtained by other authors which used both probabilistic and deterministic techniquesdemonstrates the advantages of our method.

2 Mathematical foundations

The most frequently appearing in applications are integrals with respect to a measure ofthe Gaussian type [11]. It contains various types of measures including the well- knownconditional Wiener measure. So, we consider the Lebesgue integral

JF[x]dti{x) (1)x

where F[x] is an arbitrary real measurable functional on a complete separable metric spaceX. Gaussian measure // on X is a normalized measure defined on Borel a-algebra of thespace X in the following way [10]. The value of this measure on an arbitrary cyllindricmanifold

QVl...Vn{An) = {x E X :[< ,< ip2,x >,...,< ipn,x >] E A n } ,

where (fi... cpn are the linear-independent elements of the space X' and An (n = 1, 2,...)are arbitrary Borel manifolds in Rn, is given by the formula

x / exp{--(Jr 1[u - M(

The integration in (2) is performed over the manifold of continuous functions x(t) € C[0, f3]with conditions x(0) = x,, x(/3) = Xf. It should be noted that as distinct from conven-tional path-integral approach, eq. (2) does not contain the kinetic term (the first derivativesquared) since it is included now to the measure of integration, and this circumstance sim-plifies the numerical simulations. After the appropriate change of the functional variableswe can rewrite the integral (2) with periodic boundary conditions Xj = xj = x in the formof standard conditional Wiener integral in the space Co:

Co

x{t) + xl dt}dwx. (3)

Using (3) we can compute various quantities in Euclidean quantum mechanics (or inquantum statistical mechanics accordingly). Particularly, the free energy of the system isdefined as follows:

whereoo

Z{0) = Tr exp{-/3H} = f Z(x,x,/3) dx.— oo

The ground state energy can be obtained in the following way:

= lim

We can also define the propagator

G(T) = < 0\x(0) x(r)\0 > = Jim T(T),

where the correlation function

r(r) = =1J

Co 0

T

dt]

x dwx dx.x h/^x(-)+x

The energy gap between the ground and the first excited states can be computed asfollows:

AE = E1- Eo = - lim -£- In G(T).

The ground state wave function is equal to

|*n(x)|2= lim

The generalization of eq. (2) to the case of higher dimensions is obvious.When studying the nuclear many-body problems, one needs the representation for the

many-fermion evolution operator. A number of alternative functional integral represen-tations exist for the many-fermion systems, including the many-particle generalization of

5

the path integral, an integral over an auxiliary field, integrals over the overcomplete setsof boson coherent states, determinants and Grassman variables (see [16]). Following theidea proposed in [16], in our present work we used the method based on insertion of acomplete set of antisymmetrized many particle states \x±,... xA > (A is the number ofparticles) at each time step of the discretized segment [0,/?]. As shown in [16], for theinfinitesimal time 6

X

A\ xA > —

det e x p { - l

V{\xni-xn

j

where the superscripts denote time labels and the subscripts denote particle,

X% = %i \J"n)i *n = £ Tl, I ^ 1, . . . , A.

The sign of the matrix element at each step arising from the determinant is just the signof the permutation required to bring the xn into the same order as the i""1. Hencethe cummulative sign after any number of steps is path-independent and is the signof the permutation required to make the final order equal to the initial order [16]. Itdoes not affect matrix elements of the evolution operator with antisymmetrized states orthe trace of the evolution operator times a symmetric operator (see [16]). As reportedin [4], in more than one spatial dimension, the interface between positive and negativecontributions to the functional integral degrades the statistical accuracy such that thevery good trial functions and exceedingly large ensembles are required to obtain usefulresults (it seems however that in our deterministic approach no such problems wouldarise). In one dimension the antisymmetry completely specifies the nodal points and apositive definite result may be obtained evolving the wavefunction in the ordered subspacex\ < X2 < x3 < ... < xA [4]. In this domain the determinant may be replaced by thefollowing product in the limit e —V 0 [16]:

det e x p { - l [(a? -

A

i=2

/n ( i_IIV

1 (Tn _ n \(rn-l _ n-\Xi Xi_1)\Xi xi-l£

n-l

so that the matrix element Z(x,x,/3) can be written as follows [17],[4]:

P,x,P) = \imZe(x,x,P),

where

\-NA/2 f N ,xA) 1 J N-y... axA,

A N

(4)

-iE£[vn=li>j

N A r

x n n i-n=l»=2

Some results on construction of the positive-definite functional integral representation forthe fermions in dimensions d > 2 are reported in paper [18].

Fortunately, we can make a straightforward passage to the continuum limit in eq. (4).Namely, we have found, that when e —>• 0 (or N —y oo), f3 fixed, Ze(x,x, j3) —>• Z(x,x,/3),where

4-(yj3xj(t) + xj)] dt I rf^xi... d^rcA- (5)

Here CQ is a subset of the space CQ which is the direct miltiplication of A copies of thespace Co defined above. C5

4 consists of the sequences of elements x\(t), ^ ( t ) , • • •, XA(t),Xi G Co, i = 1,2,..., A such that y/j3 xi+1(t) + xi+1 > y/j3' Xi{t) + x{ for t G [0,/3],i = 1,2,... ,A — 1. The details can be found in the Appendix.

3 Outline of the numerical method

3.1 Single functional integrals

3.1.1 Arbitrary Gaussian measure

Let % be a Hilbert space which is dence almost everywhere in X and is generated bythe measure JJL. Let {e^^i be an orthonormal basis and (•, •) be a scalar product in %.Under the following conditions on a function p(r): R 1—y X

p(r) = - p ( - r ) ;

r],p(r) > dv{r) =R

we have found that the approximation formula [12]

j F[x]dfi{x) = (2n)-n/2 f {X Rn

Rm

is exact for every polynomial functional of degree < 2m + 1. Here

k=l

k=l

wheren

^ * , x)ek,k=i

[ckm)]2 are the roots of Qm(z) = EJJL^-l)* z

m~k/k\, z e R and the measure u(v) in Rm

is a Cartesian product of symmetric normalized measures v in Euclidean space R.Formula (6) replaces the evaluation of functional integral by the computation of an

n+m - dimensional Riemannian one. Practical computations show that the good accuracy(0.1 per cent in the problems of statistical mechanics which we have considered [14]) canbe achieved with the minimal values of dimensions n = m = 1. In order to computethese integrals we use some deterministic algorithms normally employing Gaussian orTchebyshev quadratures. We have proved the convergence of approximations to exactvalue of the integral and estimated the remainder. Particularly, it follows from thisestimate that H^{F) = 0{n^m+1^) as n ->• oo.

3.1.2 Conditional Wiener measure

In the particular case of conditional Wiener measure the approximation formula (6) looksas follows:

1 1T~i r {

Since the functionals of the type F[x] = expjjjf y[x(t)]dt} often appear in applications,in many cases it is convenient to use the approximation formulas with exponential weight.For conditional Wiener integrals

I = j P[x]F[x]dwx,Co

with the weight functional

1

P[x] = exPy[p(t)x2(t) + q{t)x(t)] dt }, p{t), q(t) G C[0,

o

we obtained the following approximation formula [13]:

I

/ = (2n)-n/2 [W(l)] ~V2 expj I L2(t) dt }oo

xRn - 1 - 1

which is exact for every polynomial functional of degree < 2m + 1.Here

$[x] = F[Ax + a],

1 - / \Ax(t) = x(t) - — J B{8) W(s) x(s) ds,

1

W(t) = exp{ /"(I - s)B(s) ds } ,o

t s

(t) = [ L(s) ds-—^-[ B(s) W(s) [ f L(u) du] ds,

ot t

a(t) ^0 ^ 0 0

t 1

(8)

L(t) = J[B(s) W(s) H(s) - q(s)} ds, H(t) = j q(s) — ^ ds,o t ^ '

and B(s) is the solution of differential equation

(l-s)B'(s)-(l-s)B(s)-3B(s) = 2p(s), sE [0,1] (9)

with initial condition

Note that in the particular case p(t) = p = const, q(t) = q = const the Riccati equation(9) can be solved explicitly and the approximation formula (8) aquires the significantsimplification.

9

3.2 Multiple functional integrals

When performing the computations for quantum systems with many degrees of freedomone has to evaluate multiple functional integrals

F [ X I , ..., XX X

One of the means of computation of this kind of integral may be the successive employmentof some approximation formulas for the single functional integrals, e.g. formulas like (6)with respect to each variable x^, k = 1,..., m. But it turned out that the use of the formulaswith the given total degree of accuracy on the multiplication of spaces Xm guarantees thehigher efficiency of computations. It is clear that according to (10) we can consider themultiple functional integral again as a single integral over the space Xm [19]. Applyingour method to this case we found that approximation formula

/F[x]

where

i m

exp{--E(u«,u«)}x

x2 = 1 K

(11)

%=\is exact for every polynomial functional of the third total degree on X

In particular case of conditional Wiener measure we have

/ /

-i in

r L r) ^ •

where

x - ER

x

, 0,

i +\ _ / ~^ sign(w) , t < \v1 (1 — t)sign(w) , t > \v

Sn.(p(y, t)) = 2 E — sin(JTrt) sign(w) COS(JTTW),

(12)

f/n.(u(i)) = V2 V M ? — sin(j7rt) for alH = 1, 2 , . . . , m.

10

We compute the multidimensional Riemannian integrals which appear in (11) and (12),using the Korobov method.

For the multiple conditional Wiener integrals with the weight functional

n \

P[xu • • •, xn] = exp { J2 J \Pi(t) %Ut) + Qi{t) Xi(t)] dt },

we derived the following approximation formula [20]:

P[xu ...,xn] F[xu ...,xn] dtfx = exp { - - JT I (1 - s) B^s) ds }i=1 0

x (2n)-x exp { \ £ f L*(t) dt } £ j F[ai(-),...

where B^s) is the solution of the differential equation

(1 - s) B^s) - (1 - s)2 B2(s) - 3Bi(s) - 2Pi(s) = 0, s E [0,1]

and

fi(v,t) =sign(v; 1-tmin{\v\,i)

V- I0

0, t>\v

t s

1-t

ds},

Oi(t) = j Li(s) ds - —— j Bi(s) Wi(s) J Li(u) duds,0 * ^ 0 0

1

Wi(t)=exp{J(l-s)Bi(s)ds},o

t

L,(t) = J[B,(s) Wi(s) K,(s) - *(.,)] ds + c,

1 — U

and the constants c» are determined by the condition

I

du

f Li(s) ds =o

(13)

(14)

11

Approximation formula (13) is exact for any polynomial functional of the third summarydegree on CQ.

In the particular case pi{t) = Pi = const the solution of the Riccati equation (14) isthe following:

Hs) = 7 ^ { \fipictg[\fipi{l - s)} - -—-}.± s ± s

If we set also %(£) = - *if+9 x > - ̂ r2-

This model corresponds to the system of n particles which interact via centrifugal repul-sion and linear attraction forces with the coupling constants g and u respectively. It isbeing studied by many authors (see, e.g. [21]), which use the Monte Carlo method. Wecomputed the statistical sum (partition function) Z and the ground state energy Eo forthis model using our approximation method for functional integrals. The results for threeparticles and various values of the constant u are listed in Table 1, whereas those for thefixed u and various numbers of particles n are given in Table 2. The CPU time of com-putation of EQ was 11 sec per point u for n = 3. For comparison we cite the results E™

c

obtained by the Monte Carlo method [21] using 1000 points of time discretization and 100iterations. The exact values are denoted by EQ. The CPU time of our computation of £0for n = 11 was ca. 3 min on CDC-6500, whereas the computation of E™c required as longas 15 min on the same computer [21]. It is seen from the Tables that our deterministic

12

method gives better results than those obtained by the Monte Carlo algorithm which didnot even provide the agreement of E™c with E$ in the framework of the presented errorestimates. So, as distinct from the other deterministic techniques of computation of pathintegrals, our functional method works well also in a study of quantum systems with manydegrees of freedom, i.e. in a multidimensional case, and ensures the higher efficiency ofcomputations versus traditional stochastic methods.

4.2 Nuclear potential models

4.2.1 The triton problem

Let us now consider the numerical investigation of interaction of particles (nucleons) inthe nucleus of tritium. This three-body problem is of the real interest in physics (see[6],[22]). The Hamiltonian describing this system is the following:

| ^ £ (IT)

= (x\ ' x\ \x\ )mass rrii andHere x̂ = (x\ ' ,x\ \x\ ), i = 1,2,3 denote the coordinates of the particle with the

' ij — -̂ H

As shown in [22], various types of interaction potentials yield different values of the bindingenergy and even for the same potential the different methods of calculation give differentbinding energies.

We have studied the following model of triton used by the various authors:

V(r) = -51.5 expf - —) MeV, b = 1.6 F, (18)

nil = m-2 = mz = my, where mp = 938.279 MeV is the proton mass. The following valuesof the ground state energy have been obtained in this model by means of variational Evand Monte Carlo Emc methods (see [6],[21] and the references therein):

Emc = -9.77 ± 0.06 MeV

Ev = -9.42 MeV

Ev = -9A7±0AMeV

-9.99 ± 0.05 MeV < Ev < -9.75 ± 0.04 MeV

Ev = -9.78 MeV.

It is seen, that the difference between these results is larger than the presented errorestimates. Therefore, the solution of this problem by some other method is of interest forobtaining a more precise result.

We consider the problem (17)—(18) in the framework of the functional integral ap-proach. In order to compute the 9-fold functional integral Z we use our numerical tech-nique. The computations have been performed with the relative accuracy e = 0.01. Ourresult E = —9.7 MeV agrees well with the data of other authors. The CPU time was

13

about 15 min, which is less than the times reported in the other known works. It is de-sirable however to take into account the three- nucleon force as well as the more realisticpotentials like Argonne vu and the Paris ones [22] which would make a subject of ourfuture work.

4.2.2 One-dimensional nuclear model

We have studied the 1-dimensional nuclear model proposed in [16] with the parameterschosen so that to conform to the 3-dimensional case:

" Vk _._ r x2

Vx = 12, V2 = -12 , ox = 0.2, a2 = 0.8, h = m = l,

in units of length /0 = 1.89Fra and energy Eo = -^ = 11.6 MeV. Applying our numer-ical method in the case of antisymmetrized states in accordance with (5) and using theapproximation formula of the third total degree of accuracy for the multiple functionalintegrals, we have:

1 A } }2 n r - f J >• J

%=1 - l o

= E T3= exp{-U (19)

\jA~f}p{v,t)+Xi,xi+i,...,xA]dt} (20)

X

whereF[Xl,...,xA] =

o(v t ) _ i -tsign(v), t<

w T i _ /1 ' xx(t) < x2(t) < ... < xA(t)

U[Xl,...,xA\-^^ o t h e r w i s e _

Functional Q,[xi,... ,xA] is introduced so that to define the integration domain in thespace CQ-. It imposes the conditions on the minimal and maximal values of the inte-gration variable v in dependence on the given set of Xi,x2,... ,xA. We performed thecomputation of the Riemannian integrals using the Gaussian quadrature with the relativeaccuracy 0.01.

The 2N systemIn the case of a system of two nucleons (A=2), which can be identified with the deuteron,our reesult of computation of the binding energy for the model (19) was E = 2.4 MeVwhich can be compared with the experimental data Eex = 2.2 MeV and with the predic-tion of the semi-empirical mass formula [23] Ese = 3.5 MeV. The result can be consideredas satisfactory for such a simple model and it provides the basis for study of the more

14

realistic types of interaction.

The 4N systemFor the system of four nucleons (a-particle) we computed the binding energy with resultE = 27.6 MeV which is quite close to the experimental value Eex = 28.3 MeV [5]. Theprediction of the semi-empirical formula is Ese = 18.8 MeV. We compare our result withthose obtained in [16] in the framework of the same model by means of the lattice MonteCarlo simulations. Since the results of [16] are given in a graphical form, we reproducethem in Fig.l. It shows the binding energy for 4 particles in the dimensionless unitsE/E0: where Eo = ^ = 11.6 MeV, as a function of the lattice spacing e, obtained in [16]

by simulation of 104 events. ET and EN denote the trial energy and the normalizationenergy respectively, and EM are the values obtained by the Metropolis algorithm [16]. Theproblem of extrapolation of results to the continuum limit (e —> 0) has been discussed in[16] and [17] and found to be not simple enough. In contrast, in our approach we do nothave such problems since we do not introduce the lattice discretization and consider thequantities directly in continuum limit. Our functional-integral result EF/E0 is shown inFig.l at the point e = 0.

5 Conclusion

The described method of computation of functional integrals based on a rigorous definitionof measure in metric spaces has important advantages over conventional Monte Carlosimulation method. Due to high speed of convergence, the employment of this approachreplaces the evaluation of functional integrals by computation of the ordinary ones ofa low dimension thus allowing to use the more preferable deterministic algorithms incomputations and provides significant economy of computer time and memory. Thisapproach is very useful when other methods (perturbative, semi-classical approximation,etc.) cannot be applied. Our method works effectively also in a case of high dimensionswhere other deterministic methods of numerical path integration as well as finite-differencemethods usually fail. Moreover, it is of no importance for implementation of this methodwhether the interaction is pairwise or multiparticle, the function V(x) can depend on itscomponents x\,...xm arbitrarily, because there is no need to reverse densely filled high-order matrices. The advantages mentioned above make this method quite a promisingtool for solving the wide class of physical problems. We have found this approach to beconvenient for study of the complicated quantum systems [24]. Further work in this areais in progress.

6 Appendix

Here we will show in more detail the mechanism of the passage to the continuum limit ineq. (4). According to the definition of conditional Wiener integral given on the basis ofconsideration of stochastic processes and the Brownian motion of particles [25] we have:

JF[x(r)]dwx= (21)c

15

N

- o o - o o

where x(r) e C[0,/3], x(0) = x°, x{0) = xN, xj = xfa), r,- = je, (j = 1, . . . , N-l),£ = %. and

where XM{T) is a broken line which coincides with x(r) at the points xKIt is clear that the first exponent in (4) will tend to the multiplication of the Wiener

differentialsAN

as £ —>• 0 and the second exponent

^ N A

-.AN

exp{-— J2J1(X7 ~ X7^)2} ->• dwXl...dwxA

71=1 %~>j

A f>

while the determinant

SN = n n fi -n=li=2 L

tends to the unity in continuum limit £ —> 0. Indeed, representing SN in the form

n=l»=2

where ĉ 1 = exp{-^ rf r ^ 1 } , i = 2, 3 , . . . , A and

rni = ri(rn) = Xi(rn) - a?i_i(rn) > 0

since we impose the condition X\{r) < X2(T) < ... < XA{T), r e [0, /3], we find that

n < max _ p Y n / ( min\2\ < iC Ci — e X P \ 2 ^ * > ) '

where

Therefore,

0 < ĉ 1 < c™ax for n = 1 , . . . , iV; i = 2, 3 , . . . , A and c™1* ̂ 0 as £ ->• 0.

So,

(l-c,nox) < ( 1 - c " ) where cmax = exp{-— r ^ i n } , rmin = minr

16

r™n

and

n n (i - cmax) < n n a - c?) <n=li=2 n=lj=2

orN(A1] < SN < 1.

Expanding the expression in the left side, we obtain

— 1 ) Crnn.fr. ~\~ T7 A/ ( A — 1 ) | A ( A — 1 ) — 1 | Cmnrr — . . . + ( — 1 ) Cmn^ 0 as  —v oo.

Therefore jS1^ — 1| —>• 0 and SN —> 1 as Af —>• oo. Making the substitution of the func-tional variables at the conditional Wiener integral [25] so that to return to the standardspace CQ = {C[0,1]; x(0) = x(l) = 0} and passing to the normalized conditional Wienermeasure by insertion of the factor (2TT/3)~A//2 we come to the formula (5).

A c k n o w l e d g m e n t s - The author would like to thank the International Centre forTheoretical Physics, Trieste, for the hospitality.

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17

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18

E

0.05 0.1 0.15 0.2 0.25

Fig.l Binding energy of 4-particle bound state

19

TABLE 1Ground state energy of the Calogero model for n = 3, g = 1.5

0.10

0.20

0.25

0.50

EQ (this work)

1.346

2.700

3.366

6.738

E™c [21]

-

-

3.35 ±0.004

-

1.3472

2.6944

3.3680

6.7361

TABLE 2Ground state energy of the Calogero model for u = 0.25, g = 1.5

n

5

7

9

11

£?o (this work)

13.447

32.249

61.473

102.865

EZ [21]

13.37±0.04

32.34±0.09

61.31±0.10

102.31±0.14

E*o

13.4397

32.2718

61.5183

102.6028

20